Keyhole instability during laser welding and laser powder bed fusion (LPBF) can cause keyhole collapse and pore formation. Using high-speed x-ray imaging, we demonstrate that the flow vortex-induced protrusion on the rear keyhole wall is crucial in initiating keyhole instability. Applying a transverse magnetic field suppresses the keyhole instability by driving a secondary thermoelectric magnetohydrodynamics (TEMHD) flow that alters the net flow vortex. This minimizes protrusions and large-amplitude keyhole oscillations. The suppression effectiveness depends on the laser scanning direction relative to the magnetic field orientation because this controls the Seebeck effect-induced Lorentz force's direction. We show that at LPBF length scales, electromagnetic damping is weak, and for alloys with a large Seebeck coefficient, TEMHD becomes the dominant mechanism controlling flow behind the keyhole.